A closure result for globally hyperbolic spacetimes (Q6658176)

Recall the celebrated Myers theorem [\textit{S. B. Myers}, Duke Math. J. 8, 401--404 (1941; JFM 67.0673.01)] that if the Ricci curvature of a complete Riemannian \(n\)-manifold \((M,g)\) is bounded below, \(\mathrm{Ric} \geq (n-1)\lambda g\), then its diameter is bounded above by \(\pi/\sqrt{\lambda}\). A similar result due to \textit{G. Wei} and \textit{W. Wylie} [J. Differ. Geom. 83, No. 2, 377--405 (2009; Zbl 1189.53036) holds for the \(m\)-Bakry-Emery tensor of a complete Riemannian \(n\)-manifold \((M,g)\) with potential \(f\), namely, if \(\mathrm{Ric}^m_f = \mathrm{Ric} + \mathrm{Hess}f - df \otimes df\) is bounded below \(\mathrm{Ric}^m_f \geq (n+m-1)\lambda g\), then its diameter is bounded above by \(\pi/\sqrt{\lambda}\). In this article the authors establish an auxiliary general Myer-type theorem to obtain appropriate control of the diameter in order to establish the closure results of the kind mentioned in the title.\N\NGiven a globally hyperbolic spacetime \((I \times M^3, -N^2\,dt^2+g)\) we define the deceleration parameter \(q(t)\), the Hubble parameter \(\mathcal{H}(t)\) and the pressure parameter \(\mathcal{P}(t)\) as follows:\N\begin{itemize}\N\item[1.] \(q(t)\) is defined to be the supremum of \(-|V|_{tt}|V|/|V|_t^2\) over locally defined fields \(V\) about \((t,x)\) for over \(M_t = \{t\} \times M\),\N\item[2.] \(\mathcal{H}(t)^2\) is defined to be the infimum of the square for the mean curvature over \(M_t = \{t\} \times M\),\N\item[3.] \(\mathcal{P}(t)^2\) is defined to be the largest of the three suprema of \(T_{ii}\) over \(M_t\) where \(T\) is the stress-energy tensor.\N\end{itemize}\N\NThe authors first show that if there is a time \(t_0 \in I\) such that at \(t_0\) the deceleration parameter exceeds one half, the pressure parameter is bounded above by the cosmological constant \(\Lambda\), and the Hubble parameter is non-zero, then \(M^3\) is compact. Moreover, under these conditions \(M^3\) is covered by the curvature 1 3-sphere, and the square of its diameter is bounded above by \(16\pi^2(5q+2)/(3\mathcal{H}(t_0)^2(4q^2-1))\).\N\NThe authors go on to demonstrate that if the Einstein tensor \(G = \mathrm{Ric} - (s/2)(-Ndt^2+g)\) has that \(g\) is complete and for some \(t_0 \in I\) that, \(M_{t_0}\) has a convex energy functional, and the Hubble parameter at \(t_0\) is non-zero and that the Einstein tensor has\N\begin{align*}\NG_{ii}-kG_{\nu \nu}-\frac{1+k}{2}\mathrm{tr}(G) \geq 0\N\end{align*}\Nfor each \(i=1,2,3\), and some \((\sqrt{43}-3)/6 < k < 1+ \sqrt{3}\). Then \(M^3\) is compact. Moreover, \(M^3\) is covered by the constant curvature 1 3-sphere and the square of its diameter is bounded above by \(36\pi^2(3+2k-k^2)(4+3k)/(\mathcal{H}(t_0)^2(2+2k-k^2)(18k^2+18k-17))\).